(A + B) working together can finish certain piece of work in 8 days and (B + C) can finish it in 12 days. If A works for 4 days and B works for 7 days, rest work is done by C in 9 days. In how many days C will finish the work working alone?

Answer

To solve the problem step by step, let's denote the efficiencies of A, B, and C as \(a\), \(b\), and \(c\) respectively.

### Step 1: Calculate the efficiencies of A + B and B + C
Given:
- A + B can finish the work in 8 days.
- B + C can finish the work in 12 days.

The total work can be represented as 1 unit of work. Therefore, we can express the efficiencies as follows:
- Efficiency of A + B:
\[ a + b = \frac{1}{8} \quad \text{(work done in one day)} \]
- Efficiency of B + C:
\[ b + c = \frac{1}{12} \quad \text{(work done in one day)} \]

### Step 2: Find the total work done by A and B in 4 days and B in 7 days
From the problem, A works for 4 days and B works for 7 days. The work done by A and B can be calculated as:
- Work done by A in 4 days:
\[ 4a \]
- Work done by B in 7 days:
\[ 7b \]

### Step 3: Calculate the remaining work done by C
The total work is 1 unit, so the remaining work after A and B have worked is done by C in 9 days:
- Work done by C in 9 days:
\[ 9c \]

### Step 4: Set up the equation for total work
Now we can set up the equation for the total work:
\[ 4a + 7b + 9c = 1 \]

### Step 5: Substitute the values of a + b and b + c
From the previous steps, we have:
1. \(a + b = \frac{1}{8}\)
2. \(b + c = \frac{1}{12}\)

We can express \(a\) and \(c\) in terms of \(b\):
- From \(a + b = \frac{1}{8}\):
\[ a = \frac{1}{8} - b \]
- From \(b + c = \frac{1}{12}\):
\[ c = \frac{1}{12} - b \]

### Step 6: Substitute \(a\) and \(c\) into the total work equation
Substituting \(a\) and \(c\) into the total work equation:
\[ 4\left(\frac{1}{8} - b\right) + 7b + 9\left(\frac{1}{12} - b\right) = 1 \]

### Step 7: Simplify the equation
Expanding the equation:
\[ \frac{4}{8} - 4b + 7b + \frac{9}{12} - 9b = 1 \]
\[ \frac{1}{2} - 4b + 7b + \frac{3}{4} - 9b = 1 \]
Combining like terms:
\[ \frac{1}{2} + \frac{3}{4} - 6b = 1 \]
Convert \(\frac{1}{2}\) to quarters:
\[ \frac{2}{4} + \frac{3}{4} - 6b = 1 \]
\[ \frac{5}{4} - 6b = 1 \]
Subtract \(\frac{5}{4}\) from both sides:
\[ -6b = 1 - \frac{5}{4} \]
\[ -6b = \frac{4}{4} - \frac{5}{4} = -\frac{1}{4} \]
Dividing both sides by -6:
\[ b = \frac{1}{24} \]

### Step 8: Find \(a\) and \(c\)
Now substitute \(b\) back to find \(a\) and \(c\):
\[ a = \frac{1}{8} - \frac{1}{24} = \frac{3}{24} - \frac{1}{24} = \frac{2}{24} = \frac{1}{12} \]
\[ c = \frac{1}{12} - \frac{1}{24} = \frac{2}{24} - \frac{1}{24} = \frac{1}{24} \]

### Step 9: Calculate the time taken by C to finish the work alone
Since \(c = \frac{1}{24}\), this means C can finish the work in:
\[ \text{Time taken by C} = \frac{1}{c} = \frac{1}{\frac{1}{24}} = 24 \text{ days} \]

### Final Answer
C will finish the work alone in **24 days**.